To prove that
we verify conditions of the proposition
(
Convergence lemma
for family of complex numbers
) pointwise in
for
:
We substitute the Taylor decomposition of
around
in the following
form:
for some fixed
and numbers
,
.
Hence,
Using the condition 3 of the proposition we rewrite the above
as
The first term tends to zero according to the
condition
According to the
condition
we
have
hence
We use the proposition (
Lyapunov
inequality
)
for
,
and
to
conclude
or
This shows
that
because the
can be arbitrarily small. The verification of the rest is similar.